How to cut one group of faces with another group of faces?
You could try and use edit mode Boolean Operations (only available in Blender 2.77+)

Go into edit mode, select one of the groups of faces you want to use, open the Spacebar Menu and search for Boolean. Then choose one the the options that suits you best.

If you need the original intact faces you may need to create a copy of your objects first.



Have in mind that these will merge faces together and likely result in bad topology and non manifold meshes.

You should always strive for clean topology manifold meshes, and quad-based or quad-dominant topology